2012/03/31 by Klas Modin
Mathematics · Physics and Astronomy · #Differential geometry #Factorization #Geodesic #Information geometry #Metric (unit) #Nonlinear Waves and Solitons #Quantum Mechanics and Non-Hermitian Physics #Smoothness #Space (punctuation) #Statistical Mechanics and Entropy #Uniqueness #math-ph #math.AP #math.DG #math.FA #math.MP #msc:35Q31 #msc:53C21 #msc:58B20 #msc:58D05 #msc:58D15 #msc:65F99 #msc:94A17
paper · pdf · doi:10.1007/s12220-014-9469-2
published as J. Geom. Anal. (Feb 2014) · 30 pages, published in J. Geom. Anal
openalex publication_date 2014/02/19 · arxiv created 2014/02/28 · arxiv updated 2014/03/03 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We study geodesic equations for a family of right-invariant Riemannian metrics on the group of diffeomorphisms of a compact manifold. The metrics descend to Fisher's information metric on the space of smooth probability densities. The right reduced geodesic equations are higher-dimensional generalisations of the μ--Hunter--Saxton equation, used to model liquid crystals under influence of magnetic fields. Local existence and uniqueness results are established by proving smoothness of the geodesic spray. The descending property of the metrics is used to obtain a novel factorisation of diffeomorphisms. Analogous to the polar factorisation in optimal mass transport, this factorisation solves an optimal information transport problem. It can be seen as an infinite-dimensional version of QR factorisation of matrices.